Chapter 22 Materials Selection and Design Considerations

For a cylindrical
shaft of length L and
radius r that is
stressed in torsion,
expression for mass
in terms of density
and shear strength of
the shaft material
Strength performance
index expression for
a torsionally stressed
cylindrical shaft
22.2 Strength Considerations—Torsionally Stressed Shaft • W89
It is now necessary to take into consideration material mass. The mass m of any
given quantity of material is just the product of its density and volume. Since
the volume of a cylinder is just pr then
2 (r)
L,
m � pr 2 Lr
or, the radius of the shaft in terms of its mass is just
m
r �
A pLr
Substitution of this r expression into Equation 22.4 leads to
tf N �
2Mt m
p a
A pLr b
3
pL
� 2Mt B
3r3 m3 Solving this expression for the mass m yields
m � 12NM t2 2�3 1p 1�3 L2 a r
(22.5)
(22.6)
(22.7)
(22.8)
The parameters on the right-hand side of this equation are grouped into three sets of
parentheses.Those contained within the first set (i.e., N and ) relate to the safe functioning
of the shaft.Within the second parentheses is L, a geometric parameter. Finally,
the material properties of density and strength are contained within the last set.
The upshot of Equation 22.8 is that the best materials to be used for a light
shaft that can safely sustain a specified twisting moment are those having low r�t
ratios. In terms of material suitability, it is sometimes preferable to work with what
is termed a performance index, P, which is just the reciprocal of this ratio; that is,
2�3
Mt f
P � t2�3 f
r
t2�3 f
(22.9)
In this context we want to utilize a material having a large performance index.
At this point it becomes necessary to examine the performance indices of a variety
of potential materials. This procedure is expedited by the utilization of what
are termed materials selection charts. 1 These are plots of the values of one material
property versus those of another property. Both axes are scaled logarithmically and
usually span about five orders of magnitude, so as to include the properties of virtually
all materials. For example, for our problem, the chart of interest is logarithm
of strength versus logarithm of density, which is shown in Figure 22.2. 2 It may be
noted on this plot that materials of a particular type (e.g., woods, engineering polymers,
etc.) cluster together and are enclosed within an envelope delineated with a
bold line. Subclasses within these clusters are enclosed using finer lines.
1 A comprehensive collection of these charts may be found in M. F. Ashby, Materials Selection
in Mechanical Design, 2nd edition, Butterworth-Heinemann, Woburn, UK, 2002.
2 Strength for metals and polymers is taken as yield strength, for ceramics and glasses, compressive
strength, for elastomers, tear strength, and for composites, tensile failure strength.
b